Gyration Stability for Projective Planes
Sebastian Chenery, Stephen Theriault
TL;DR
This work presents a nearly complete homotopy-theoretic classification of gyrations on the complex, quaternionic, and octonionic projective planes. By modeling a k-gyration as a pushout and analyzing the top-cell attaching map φτ, the authors derive a concrete obstruction δ governed by the J-homomorphism, Toda’s homotopy groups, and Whitehead products. A central lambda_delta framework reduces stability questions to the existence of a λ∈π_{m+k−1}(S^m) satisfying a congruence f∘arτ+λ∘Σ^{k−1}f+[ι_m,λ]≃±f∘arω, enabling explicit computations across k in {2,3,4,5,6,7,8,9,10,12,13,14}. They obtain precise G^k-stability and GSII-type counts: ℂℙ^2 is G^2-stable; ℍℙ^2 is not G^2-stable but G^4-stable (and G^3, G^5, G^6 stable); for 𝕆ℙ^2, stability fails at k=2 and k=4 but holds at k=8,9,10 while failing at k=12, with higher k exhibiting a rich array of homotopy types guided by Toda- and J-theory. The results illuminate how twistings interact with the J-image to govern the homotopy types of gyrations in projective planes, advancing the understanding of gyration stability in geometric topology.
Abstract
Gyrations are operations on manifolds that arise in geometric topology, where a manifold $M$ may exhibit distinct gyrations depending on the chosen twisting. For a given $M$, we ask a natural question: do all gyrations of $M$ share the same homotopy type regardless of the twisting? A manifold with this property is said to have gyration stability. Inspired by recent work by Duan, which demonstrated that the quaternionic projective plane is not gyration stable with respect to diffeomorphism, we explore this question for projective planes in general. We obtain a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes up to homotopy.